Can I find a mapping that minimizes the maximum distance ratio of certain vectors?
An interesting question. I would start by finding n nearest neighbors of each data point, then calculate their center of mass c and the point's distance d to its nth nearest neighbor. The smaller the d is the larger the density is around a given point. You could then iteratively step every point away from their c in an inverse proportion to the distance d with a suitable step size. This would spread out the clusters.

But this won't help you transform any new points in the dataset, maybe you can learn this arbitrary mapping R^n -&gt; R^n by using an other neural network and apply it to new samples?

This is the first ad-hoc idea which came to my mind. It would be interesting to see a 2D animation of this.

A more rigorous approach might be a variational autoencoder, you can embed the data in a lower dimensional space with approximately normal distribution. But it doesn't guarantee that clusters would be as spread out as you'd like. An alternative loss function would help with that, for example every point's distance to their original nth closest neighbor should be as close to one as possible.